The Mattila-Sj\"olin problem for the k-distance over a finite field

TL;DR

This paper investigates the Mattila-Sj"olin problem for a $k$-norm variant over finite fields, establishing bounds on the size of subsets needed for their distance sets to cover the entire field, with results matching known bounds in certain cases.

Contribution

It extends the Mattila-Sj"olin problem to a $k$-norm setting over finite fields, providing sharp results in odd dimensions and using combinatorial Fourier analysis techniques.

Findings

For some $k$, the $k$-norm distance set covers the entire field with subsets of size $Cq^rac{d+1}{2}$.

The results match those of Iosevich and Rudnev (2007) for the classical norm.

The bounds are sharp in all odd dimensions.

Abstract

Let $\mathbb{F}_q^d$ be a $d$-dimensional vector space over a finite field $\mathbb{F}_q$ with $q$ elements. For $x\in \mathbb{F}_q^d$, let $\|x\| = x_1^2+\dots+x_d^2$. By abuse of terminology, we shall call $\|\cdot\|$ a norm on $\mathbb{F}_q^d$. For a subset $E\subset \mathbb{F}_q^d$, let $\Delta(E)$ be the distance set on $E$ defined as $\Delta(E):=\{\|x-y\| : x, y \in E \}$. The Mattila-Sj\"olin problem seeks the smallest exponent $\alpha>0$ such that $\Delta(E) =\mathbb{F}_q$ for all subsets $E \subset \mathbb{F}_q^d$ with $|E| \geq Cq^\alpha$. In this article, we consider this problem for a variant of this norm, which generates a smaller distance set than the norm $\|\cdot\|.$ Namely, we replace the norm $\|\cdot\|$ by the so-called $k$-norm $(1 \leq k \leq d)$, which can be viewed as a kind of deformation of $\|\cdot\|$. To derive our result on the Mattila-Sj\"olin problem for the $k$-norm, we use a combinatorial method to analyze various summations arising from the discrete Fourier machinery. Even though our distance set is smaller than the one in the Mattila-Sj\"olin problem, for some $k$ we still obtain the same result as that of Iosevich and Rudnev (2007), which deals with the Mattila-Sj\"olin problem. Furthermore, our result is sharp in all odd dimensions.